1-1 Points, Lines, and Planes
Refer to the figure.
13. Name the lines that are only in plane Q.
SOLUTION: There are only two lines that lie in plane Q: n and q.
14. How many planes are labeled in the figure?
SOLUTION: There are two planes labeled in the figure, namely Q and R.
15. Name the plane containing the lines m and t.
SOLUTION: A plane is a flat surface made up of points that extends infinitely in all directions. Here, the plane containing the lines
m and t is R.
16. Name the intersection of lines m and t.
SOLUTION: The two lines m and t intersect at the point C on the plane R.
17. Name a point that is not coplanar with points A, B, and C.
SOLUTION: Coplanar points are points that lie in the same plane. Here, the points A, B, and C lie on the plane R. So, to find a
point which is NOT coplanar with A, B, and C, consider any point on the plane Q. The point P, which is on the plane
Q, is not coplanar with the points A, B, and C.
18. Are points F, M , G, and P coplanar? Explain.
SOLUTION: Coplanar points are points that lie in the same plane. Here, the points G and P lie on the plane Q. But the point M lies
between the planes Q and R and the point F lies on the plane R.
19. Name the points not contained in a line shown.
SOLUTION: The points A and P do not lie in any of the lines shown on the planes Q and R.
20. What is another name for line t?
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SOLUTION: There are two points C and E marked on the line t. So, the line t can also be named as
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between the planes Q and R and the point F lies on the plane R.
19. Name the points not contained in a line shown.
SOLUTION: 1-1 Points,
Lines, and Planes
The points A and P do not lie in any of the lines shown on the planes Q and R.
20. What is another name for line t?
SOLUTION: There are two points C and E marked on the line t. So, the line t can also be named as
21. Does line n intersect line q? Explain.
SOLUTION: A line and a plane can be extended infinitely. Lines n and q are coplanar but not parallel. So, when the lines n and q
are extended on the plane Q, they will intersect.
Name the geometric term(s) modeled by each object.
22. SOLUTION: The tip of a pen denotes a location. So, it models a point.
23. SOLUTION: The roof segment is a flat surface that extends in all directions, so it is a plane. The edges of a roof model lines,
which intersect at the corners. So, the highlighted parts model intersecting lines.
24. SOLUTION: The chessboard is a flat surface that extends in all directions. So, it is a plane. Also it has lines that intersect on the
plane. So, it also models intersecting lines.
25. SOLUTION: eSolutions
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In the figure two flat surfaces intersect each other. The two flat surfaces model two planes, so the whole figure
models two planes intersecting to form a line.
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24. SOLUTION: The chessboard
is a Planes
flat surface that extends in all directions. So, it is a plane. Also it has lines that intersect on the
1-1 Points,
Lines, and
plane. So, it also models intersecting lines.
25. SOLUTION: In the figure two flat surfaces intersect each other. The two flat surfaces model two planes, so the whole figure
models two planes intersecting to form a line.
26. a blanket
SOLUTION: A blanket is a flat surface that extends in all directions. So, it models a plane.
27. a knot in a rope
SOLUTION: A knot in a rope denotes a location. So, it models a point.
28. a telephone pole
SOLUTION: A telephone pole models a line.
29. the edge of a desk
SOLUTION: The edge of a desk models a line.
30. two connected walls
SOLUTION: Each wall models a plane. Two connected walls model intersecting planes.
31. a partially opened folder
SOLUTION: Each side of a folder is a flat surface that extends in all directions. So, a partially opened folder models two
intersecting planes.
Draw and label a figure for each relationship.
32. Line m intersects plane R at a single point.
SOLUTION: Draw a plane R and add only one point. Draw line m vertically through the point. Dash the line to indicate the portion
hidden by the plane. eSolutions Manual - Powered by Cognero
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31. a partially opened folder
SOLUTION: Each side
of a folder
is a flat surface that extends in all directions. So, a partially opened folder models two
1-1 Points,
Lines,
and Planes
intersecting planes.
Draw and label a figure for each relationship.
32. Line m intersects plane R at a single point.
SOLUTION: Draw a plane R and add only one point. Draw line m vertically through the point. Dash the line to indicate the portion
hidden by the plane. 33. Two planes do not intersect.
SOLUTION: Draw two planes that do not intersect, that is, two planes that are parallel.
34. Points X and Y lie on
SOLUTION: Draw a line
and plot two points X and Y on the line.
35. Three lines intersect at point J but do not all lie in the same plane.
SOLUTION: Draw two lines that intersect at a point on a plane. Then draw a line perpendicular to the plane so the line does not
lie in the same plane. Dash the portion of the vertical line to indicate that it is hidden by the plane. eSolutions Manual - Powered by Cognero
36. Points A(2, 3), B(2, –3), C and D are collinear, but A, B, C, D, and F are not.
SOLUTION: Page 4
SOLUTION: Draw a line
and plot two points X and Y on the line.
1-1 Points, Lines, and Planes
35. Three lines intersect at point J but do not all lie in the same plane.
SOLUTION: Draw two lines that intersect at a point on a plane. Then draw a line perpendicular to the plane so the line does not
lie in the same plane. Dash the portion of the vertical line to indicate that it is hidden by the plane. 36. Points A(2, 3), B(2, –3), C and D are collinear, but A, B, C, D, and F are not.
SOLUTION: First, plot the point A(2, 3), and B(2, –3) on a coordinate plane. Now plot the points C and D on the same line as A
and B. Plot another point F which is not in the same line as A and B.
37. Lines
and are coplanar but do not intersect.
SOLUTION: Draw two parallel lines
38. and
and on a plane.
intersect at P(4, 3), where point F is at ( –2, 5) and point J is at (7, 9).
SOLUTION: Plot and label the points P(4, 3), F( –2, 5), and J(7, 9) on a coordinate plane. Draw a straight line through points P
and J. Similarly, draw a straight line through points P and F.
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1-1 Points, Lines, and Planes
38. and
intersect at P(4, 3), where point F is at ( –2, 5) and point J is at (7, 9).
SOLUTION: Plot and label the points P(4, 3), F( –2, 5), and J(7, 9) on a coordinate plane. Draw a straight line through points P
and J. Similarly, draw a straight line through points P and F.
39. Lines s and t intersect, and line v does not intersect either one.
SOLUTION: Draw two parallel lines t and v on a coordinate plane. Then draw a line s perpendicular to both the plane and the line
t, but does not intersect the line v. Dash the portion of line s to indicate it is hidden by the plane. PACKING When packing breakable objects such as glasses, movers frequently use boxes with inserted
dividers like the one shown.
40. How many planes are modeled in the picture?
SOLUTION: There are 5 planes that give the box the shape of a rectangular prism: the bottom and four sides. Then there are 4
more planes modeled by the partially opened top flaps. Inside the box, the space is divided by 6 planes which
intersect each other. So, there are a total of 15 planes in the figure.
41. What parts of the box model lines?
SOLUTION: The edges of the sides of the box model lines. The dividers represent planes and the intersection of planes are lines.
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So, the edges of the dividers also model lines. 42. What parts of the box model points?
SOLUTION: There are 5 planes that give the box the shape of a rectangular prism: the bottom and four sides. Then there are 4
more planes modeled by the partially opened top flaps. Inside the box, the space is divided by 6 planes which
1-1 Points,
and Planes
intersectLines,
each other.
So, there are a total of 15 planes in the figure.
41. What parts of the box model lines?
SOLUTION: The edges of the sides of the box model lines. The dividers represent planes and the intersection of planes are lines.
So, the edges of the dividers also model lines. 42. What parts of the box model points?
SOLUTION: Each vertex of the box is a location which models a point.
Refer to the figure below.
43. Name two collinear points.
SOLUTION: Collinear points are points that lie on the same line. Here, points M and N lie on the same line, so they are collinear
points. (Note that there are many other pairs of collinear points.)
44. How many planes appear in the figure?
SOLUTION: A plane is a flat surface that extends infinitely in all directions. Here, the 5 rectangular sides of the figure represent 5
planes, the top pentagonal face represents another plane. Since the base of the prism lies on plane A, it only
represents one additional plane, the seventh plane. So, 7 planes appear in the figure.
45. Do plane A and plane MNP intersect? Explain.
SOLUTION: MNP is the top face of the solid, and does not have any common lines with the plane A. So, they do not intersect.
46. In what line do planes A and QRV intersect?
SOLUTION: The plane QRV contains the rectangle QRVN. This rectangle intersects the plane A in the line
47. Are points T, S, R, Q, and V coplanar? Explain.
SOLUTION: Coplanar points are points that lie in the same plane. Here, the points T, S, R, and Q all lie on the plane A; there is no
other plane which contains all four of them. But the point V does not lie on plane A. Therefore, they are not coplanar.
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